Geometry of odd crown blocks

Abstract

Parity imbalance determines extremality of each odd block.

The parity-majority orientation is the odd-block argument in the proof of source Lemma 3.2 and the endpoint selection in Proposition 3.3.

Theorem 1.1 (Odd blocks have a unique parity majority).

Lean statement: D5/S3/Combinatorics/Geometry/CrownOrderPolytopeOddBlocks.crownOddBlock_geometry

Proof. Machine-checked in Lean as D5/S3/Combinatorics/Geometry/CrownOrderPolytopeOddBlocks.crownOddBlock_geometry (✓ std3). ∎

Citation. Teemu Lundström and Leonardo Saud Maia Leite (2025). Order polytopes of crown posets. DOI: 10.48550/arXiv.2504.05123. URL: https://arxiv.org/abs/2504.05123v3.

Commentary.

For n at least two, every odd block of a connected cycle partition has even-vertex and odd-vertex counts differing by one. An even majority excludes incoming crown edges from other blocks; an odd majority excludes outgoing edges to other blocks. These orientations are conclusions derived from the actual block geometry.

References

• Truth anchor: D5/S3/Combinatorics/Geometry/CrownOrderPolytopeOddBlocks.crownOddBlock_geometry
• Dependency: D5/S3/Combinatorics/Geometry/CrownOrderPolytopeCyclePartitions